Equilibrium Stat Mech Vs. Kinetics

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Ensembles

Figured it out thanks, next post:

 

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Related to this point (functioning as something which might explain the above more deeply), what is the actual meaning of this passage from Landau vol. 5:

Let us now consider a macroscopic body or system of bodies, and assume that the system is closed, i.e. does not interact with any other bodies. A part of the system, which is very small compared with the whole system but still macroscopic, may be imagined to be separated from the rest; clearly, when the number of particles in the whole system is sufficiently large, the number in a small part of it may still be very large. Such relatively small but still macroscopic parts will be called subsystems. A subsystem is again a mechanical system, but not a closed one; on the contrary, it interacts in various ways with the other parts of the system. Because of the very large number of degrees of freedom of the other parts, these interactions will be very complex and intricate. Thus the state of the subsystem considered will vary with time in a very complex and intricate manner.

An exact solution for the behaviour of the subsystem can be obtained only by solving the mechanical problem for the entire closed system, i.e. by setting up and solving all the differential equations of motion with given initial conditions, which, as already mentioned, is an impracticable task. Fortunately, it is just this very complicated manner of variation of the state of subsystems which, though rendering the methods of mechanics inapplicable, allows a different approach to the solution of the problem.

A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state. This may be more precisely formulated as follows. Let dp dq denote some small "volume" of the phase space of the subsystem, corresponding to coordinates q, and momenta p, lying in short intervals dq, and dp,. We can say that, in a sufficiently long time T, the extremely intricate phase trajectory passes many times through each such volume of phase space. Let dt be the part of the total time T during which the subsystem was in the given volume of phase space dp dq. When the total time T increases indefinitely, the ratio dt/T tends to some limit w = lim dt/T.
This quantity may clearly be regarded as the probability that, if the subsystem is observed at an arbitrary instant, it will be found in the given volume of phase space dpdq.

Again as an example, take H(q,p) = \tfrac{p^2}{2m}+\tfrac{k}{2}q^2 = E_0 as the Hamiltonian for a single particle, the trajectory of the particle, i.e. the set of all possible states, is an ellipse in (q,p) phase space.

Extend it to n particles and we'll have a Cartesian product of n ellipses, or a big ellipsoid, visualized as

What in the world does that passage really mean, in terms of n harmonic oscillators respresented as a single curve in phase space? I can't make actually make any sense out of it.

 

[TSC]

This is the ergodic hypothesis.